It will be recalled that channel “block encoding” consists, when the “codewords” sent to a receiver or recorded on a data carrier are formed, of introducing a certain level of redundancy in the data. More particularly, by means of each codeword, the information is transmitted that is initially contained in a predetermined number k of symbols taken from an “alphabet” of finite size q; on the basis of these k information symbols, calculation is made of a number n>k of symbols belonging to that alphabet, which constitute the components of the codewords: v=(v1, v2, . . . , vn). The set of codewords obtained when each information symbol takes some value in the alphabet constitutes a sort of dictionary referred to as a “code” of “dimension” k and “length” n.
When the size q of the alphabet is a power of a prime number, the alphabet can be given a field structure known as a “Galois field” denoted Fq, of which the non-zero elements may conveniently be identified as each being equal to γi for a corresponding value of i, where i=1, . . . , q−1, and where γ is a primitive (q−1)th root of unity in Fq.
In particular, certain codes, termed “linear codes” are such that any linear combination of codewords (with the coefficients taken from the alphabet) is still a codeword. These codes may conveniently be associated with a matrix H of dimension (n−k)×n, termed “parity-check matrix”: a word v of given length n is a codeword if, and only if, it satisfies the relationship: H·vT=0 (where the exponent T indicates the transposition); the code is then said to be “orthogonal” to the matrix H.
At the receiver, the associated decoding method then judiciously uses this redundancy to detect any transmission errors and if possible to correct them. There is a transmission error if the difference e between a received word r and the corresponding codeword v sent by the transmitter is non-zero.
More particularly, the decoding is carried out in two main steps.
The first step consists of associating an “associated codeword” with the received word. To do this, the decoder first of all calculates the “error syndromes vector” s=H·rT=H·eT of length (n−k) (in the context of the present invention, no difference is made between the term “word” and the term “vector”). If the syndromes are all zero, it is assumed that no transmission error has occurred, and the “associated codeword” will then simply be taken to be equal to the received word. If that is not the case, it is thereby deduced that the received word is erroneous, and calculations are then performed that are adapted to estimate the value of the error e; in other words, these calculations provide an estimated value {circumflex over (e)} such that (r−{circumflex over (e)}) is a codeword, which will then constitute the “associated codeword”. Usually, this first step of the decoding is divided into two distinct sub-steps: a first so-called “error locating” sub-step, during which the components of the received word are determined of which the value is erroneous, and a second so-called “error correction” sub-step, during which an estimation is calculated of the transmission error affecting those components.
The second step simply consists in reversing the encoding method. In the ideal situation in which all the transmission errors have been corrected, the initial information symbols are thereby recovered.
It will be noted that in the context of the present invention, reference will often be made to “decoding” for brevity, to designate solely the first of those steps, it being understood that the person skilled in the art is capable without difficulty of implementing the second step.
The objective usually given to the decoding is to associate with the received word the codeword situated at the shortest Hamming distance from this received word, the “Hamming distance” being, by definition, the number of places where two words of the same length have a different symbol. The shortest Hamming distance between two different codewords of a code is termed the “minimum distance” d of that code. This is an important parameter of the code. More particularly, it is in principle possible to find the position of the possible errors in a received word, and to provide the correct replacement symbol (i.e. that is identical to that sent by the transmitter) for each of those positions, each time the number of erroneous positions is at most equal to INT[(d−1)/2] (where “INT” designates the integer part) for a code of minimum distance d (for certain error configurations, it is sometimes even possible to achieve better). However, in all cases, the concern is not with a possibility in principle, since it is often difficult to develop a decoding algorithm achieving such performance. It should also be noted that, when the chosen algorithm manages to propose a correction for the received word, that correction is all the more reliable (at least, for most transmission channels) the smaller the number of positions it concerns.
Among known codes, “Reed-Solomon” codes may be cited, which are reputed for their efficiency. They are linear codes, of which the minimum distance d is equal to (n−k+1). The parity-check matrix H of the Reed-Solomon code of dimension k and length n (where n is necessarily equal to (q−1) or to a divisor of (q−1)) is a matrix with (n−k) lines and n columns, which has the structure of a Vandermonde matrix. This parity-check matrix H, which may be defined for example by taking Hij=αi(j−1) (1≦i≦n−k, 1≦j≦n), where α is an nth root of unity in Fq; it is then possible to label the component vj, where 1≦j≦n, of any codeword v=(v1, v2, . . . , vn) by means of the element α(j−1) of Fq; it is for this reason that a set such as (1,α,α2, . . . , αn−1) is termed “locating set” of the Reed-Solomon code.
As mentioned above, the step of a method of decoding during which a “codeword associated with the received word” is calculated is usually divided into two sub-steps: the first sub-step referred to as an “error locating” sub-step, consists of identifying in the received word the components whose value is erroneous; and the second sub-step consists then of calculating the corrected value of those erroneous components.
For the decoding of Reed-Solomon codes, as regards error locating, use is usually made of the algorithm known as the “Berlekamp-Massey” algorithm, which will now be briefly described: firstly a matrix S is constructed, termed “syndromes matrix”, of which each element is a certain component of the error syndromes vectors s=H·rT=H·eT; next a vector Λ is sought such that Λ·S=0, then a polynomial Λ(Z) is formed of which the coefficients are components of the vector Λ; the inverses of the roots of that polynomial Λ(Z) are then, among the elements ωi (where i=1, . . . , n) of the locating set, those which label the erroneous components of the received word r.
As regards the error correction, use is usually made of the algorithm known as the “Forney” algorithm which will now be briefly described. The error calculating polynomial Ω(Z)=Λ(Z)S(Z) modulo Zn−k is constructed, where
      S    ⁡          (      Z      )        =            ∑              i        =        0                    n        -        k        -        1              ⁢                  ⁢                  s        i            ⁢              Z        i            and the si are the components of the error syndromes vector s; the errors are then given, for i=1, . . . , n, by:
      e    i    =      {                                        0                                if                                                              Λ                ⁡                                  (                                      ω                    i                                          -                      1                                                        )                                            ≠              0                                                                          -                                                Ω                  ⁢                                                                          ⁢                                      (                                          ω                      i                                              -                        1                                                              )                                                                                        p                    i                                    ⁢                                                            Λ                      ′                                        ⁡                                          (                                              ω                        i                                                  -                          1                                                                    )                                                                                                                if                                                              Λ                ⁡                                  (                                      ω                    i                                          -                      1                                                        )                                            =              0                                          ,      where Λ′(Z) designates the derivative of Λ(Z), and pi is equal to 1 for a “standard” Reed-Solomon code and at the diagonal element in position (i,i) of the matrix P for a modified code (see below).
For more details on Reed-Solomon codes, and in particular the algorithms of Berlekamp-Massey and of Forney, reference may for example be made to the work by R. E. Blahut entitled “Theory and practice of error-control codes”, Addison-Wesley, Reading, Mass., 1983.
For modern information carriers, for example on computer hard disks, CDs (“compact discs”) and DVDs (“digital video discs”), it is sought to increase the density of information. When such a carrier is affected by a physical defect such as a scratch, a high number of information symbols may be rendered unreadable. This problem may nevertheless be remedied by using a very long code. However, as indicated above, the length n of the words in Reed-Solomon codes is less than the size q of the alphabet of the symbols. Consequently, if a Reed-Solomon code is desired having codewords of great length, high values of q must be envisaged, which leads to costly implementations in terms of calculation and storage in memory. Moreover, high values of q are sometimes ill-adapted to the technical application envisaged. For this reason, it has been sought to build codes which naturally provide words of greater length than Reed-Solomon codes without however requiring a longer alphabet.
In particular so-called “algebraic geometric codes” or “Goppa geometric codes” have recently been proposed (see for example the article by Tom Høholdt and Ruud Pellikaan entitled “On the Decoding of Algebraic-Geometric Codes”, IEEE Trans. Inform. Theory, vol. 41 n° 6, pages 1589 to 1614, November 1995). These codes are constructed from a set of n pairs (x,y) of symbols belonging to a chosen Galois field Fq; this set of pairs constitutes the locating set of the algebraic geometric code. In general terms, there is an algebraic equation with two unknowns X and Y such that the pairs (x,y) of that locating set are all solutions of that algebraic equation. The values of x and y of these pairs may be considered as coordinates of “points” Pβ (where β=1, . . . , n) forming an “algebraic curve”.
An important parameter of such a curve is its “genus” g. In the particular case where the curve is a simple straight line (the genus g is then zero), the algebraic geometric code reduces to a Reed-Solomon code. For given q and g, certain algebraic curves, termed “maximum”, make it possible to achieve a length equal to (q+2 g√{square root over (q)}), which may be very high; for example, with an alphabet size of 256 and a genus equal to 120, codewords are obtained of length 4096.
In the context of the present invention, a very general class of algebraic geometric codes is concerned: these codes, of which an example described in detail will be found below, are defined on an algebraic curve represented by an equation f(X, Y)=0 withf(X,Y)=Xb+cYa+ΣcijYjXi,where c≠0 and the cij are elements of Fq, a and b are strictly positive mutually prime integers, and where the sum only applies to the integers i and j which satisfy ai+bj<ab. This form of equation is referred to as “C(a,b)”. For such a code, a parity-check matrix is conventionally defined in the following manner. With every monomial YjXi, where i and j are positive integers or zero, a “weight” is associated (see below for details). If, for an integer ρ≧0, there is at least one monomial of which the weight is ρ, it is said that ρ is an “achievable” weight. Let ρ1<ρ2< . . . <ρn−k be the (n−k) smallest achievable weights, and let hα (where α=1, . . . , n−k) be a monomial of weight ρα. The element in line α and column β of the parity-check matrix is equal to the monomial hα evaluated at the point Pβ (where, it may be recalled, β=1, . . . , n) of the algebraic curve. Each point Pβ then serves to identify the βth component of any codeword. A code having such a parity-check matrix is termed a “one-point” code since its parity-check matrix is obtained by evaluating (at the n points Pβ) functions (the monomials hα) which have poles only at a single point, i.e. the point at infinity.
Like all codes, algebraic geometric codes may be “modified” and/or “shortened”. It is said that a given code Cmod is a “modified” version of the code C if there is a square non-singular diagonal matrix A such that each word of Cmod is equal to v·A with v being in C. It is said that a given code is a “shortened” version of the code C if it comprises solely the words of C of which, for a number R of predetermined positions, the components are all zero: as these positions are known to the receiver, their transmission can be obviated, such that the length of the shortened code is (n−R). In particular, it is common to shorten an algebraic geometric code by removing from the locating set, where possible, one or more points for which the x coordinate is zero.
Algebraic geometric codes are advantageous as to their minimum distance d, which is at least equal to (n−k+1−g), and, as has been said, as to the length of the codewords, but they have the drawback of requiring decoding algorithms that are rather complex, and thus rather expensive in terms of equipment (software and/or hardware) and processing time. This complexity is in fact greater or lesser according to the algorithm considered, a greater complexity being in principle the price to pay for increasing the error correction capability of the decoder (see for example the article by Tom Høholdt and Ruud Pellikaan cited above). Generally, the higher the genus g of the algebraic curve used, the greater the length of the codewords, but also the greater the complexity of the decoding.
Various error locating algorithms are known for algebraic geometric codes (defined on a curve of non-zero genus).
Such an algorithm, termed “basic” algorithm, has been proposed by A. N. Skorobogatov and S. G. Vladut in the article entitled “On the Decoding of Algebraic-Geometric Codes”, IEEE Trans. Inform. Theory, vol. 36 no. 5, pages 1051 to 1060, November 1990). Skorobogatov and Vladut have also proposed, in the same article cited above, a “modified” version of the “basic” algorithm, which generally enables a higher number of errors to be corrected that the “basic” algorithm.
Algorithms are also known which operate using an iterative principle: each new iteration of such an algorithm invokes a supplementary component of the syndromes vectors s=H·rT.
An example of such an iterative decoding algorithm is disclosed in the article by M. Sakata et al. entitled “Generalized Berlekamp-Massey Decoding of Algebraic-Geometric Codes up to Half the Feng-Rao Bound” (IEEE Trans. Inform. Theory, vol 41, pages 1762 to 1768, November 1995) This algorithm can be viewed as a generalization of the Berlekamp-Massey algorithm to algebraic geometric codes defined on a curve of non-zero genus.
Another example of an iterative decoding algorithm has been disclosed by M. O'Sullivan in the article “A Generalization of the Berlekamp-Massey-Sakata Algorithm” (preprint 2001).
All the error locating algorithms mentioned above provide “error locating polynomials” Λ(x,y) of which the zeros comprise all the pairs (x,y) labeling the components of the received word having suffered a transmission error. The set of the error locating polynomials forms an ideal, in which a basis known as a “Gröbner basis” can be defined.
The calculation of errors for algebraic geometric codes is prima facie more complicated than for Reed-Solomon codes. Thus:                the error locating sub-step not only produces one error locating polynomial (denoted above Λ(Z) for Reed-Solomon codes), but several polynomials, (belonging to the ideal of the error locating polynomials);        these error locating polynomials are polynomials with two variables instead of one; and        these error locating polynomials thus possess partial derivatives with respect to those two variables, such that the Forney formula given above, which involves a single derivative, is no longer applicable.        
Various error calculating algorithms are known for algebraic geometric codes.
The article “Algebraic Geometry Codes”, by Tom Høholdt, Jacobus Van Lint and Ruud Pelikaan (Chapter 10 of the “Handbook of Coding Theory”, North Holland, 1998) constructs the product of certain powers of the polynomials of the Gröbner basis. It then performs a linear combination of those products, allocated with appropriate coefficients. Finally it shows that the value of the polynomial so obtained, taken at the point (x,y) of the locating set is, with the sign being the only difference, the value of the error for the component of the received word labeled by that point (x, y).
The article “A Generalized Forney Formula for Algebraic Geometric Codes” by Douglas A. Leonard (IEEE Trans. Inform. Theory, vol. 42, n° 4, pages 1263 to 1268, July 1996), and the article “A Key Equation and the Computation of Error Values for Codes from Order Domains” by John B. Little (published on the Internet on Apr. 7, 2003) calculate the values of the errors by evaluating a polynomial with two variables at the common zeros of the error locating polynomials.
These algorithms are complex to implement, in particular due to the fact that they comprise the multiplication of polynomials with two variables, in addition to formal multiplications in Fq.
U.S. patent application Ser. No. 10/746,144, which is incorporated herein by reference, describes a decoding method which performs both the location and the correction of errors. This decoding method applies to a vast set of codes, which include in particular the one-point algebraic geometric codes, described above, defined on an algebraic curve of type C(a,b). This method will now be described in some detail.
This decoding method relies on the subdivision of the locating set of the code into subsets which we will term “aggregates”. By definition, an “aggregate” groups together the pairs (x,y) of the locating set having a common value of x (it would have been equally possible to define the aggregates with a common value of y by swapping the roles of the unknowns X and Y of the equation representing the algebraic curve on which the code is defined). When it is desired to emphasize this aggregate structure, the pairs of the locating set (which is furthermore not necessarily a set of solutions to an algebraic equation of type C(a,b)) will be denoted (x,yp(x)), where p=0, . . . ,λ(x)−1 and λ(x) is the cardinal of the aggregate considered, and the components of any word c of length n will be denoted c(x,yp(x)); the components of c which, labeled in this manner, possess the same value of x, will be said to form an “aggregate of components” of the word c.
Let m be the maximum weight of the monomials defining the lines of the parity-check matrix (see above). According to application Ser. No. 10/746,144, these monomials are classified in sets of monomialsMj={YjXi|0≦i≦(m−bj)/a}for 0≦j≦jmax, where jmax<a . The cardinal of this set Mj is thus:t(j)=1+INT[(m−bj)/a]. 
Let x1, x2, . . . , xμ denote the different values of x in the locating set, andv=[v(x1, y0(x1)), . . . , v(x1,yλ1−1(x1)), . . . ,v(xμ,yλμ−1(xμ))],denote any particular codeword for each aggregate attached to one of the values x1, x2, . . . , xμ of x, there are constructed (jmax+1) “aggregate symbols”
            v      j        ⁡          (      x      )        ≡            ∑              p        =        0                              λ          ⁡                      (            x            )                          -        1              ⁢                  ⁢                            [                                    y              p                        ⁡                          (              x              )                                ]                j            ⁢              v        ⁡                  (                      x            ,                                          y                p                            ⁡                              (                x                )                                              )                    for j=0, . . . , jmax. These aggregate symbols serve to form (jmax+1) “aggregate words”vj≡[vj(x1),vj(x2), . . . ,vj(xμ)],of length μ.
It is easily verified that the condition of belonging to the algebraic geometric code (i.e. H·vT=0) is equivalent to the set of (jmax+1) equations:Ht(j)·vjT=0,where the function t(j) is given above and is, by definition,
      H    t    =            [                                    1                                1                                ⋯                                1                                                              x              1                                                          x              2                                            ⋯                                              x              μ                                                            ⋮                                ⋮                                ⋮                                ⋮                                                              x              1                              t                -                1                                                                        x              2                              t                -                1                                                          ⋯                                              x              μ                              t                -                1                                                        ]        .  
The advantage of this formulation is that the matrix Ht of the equation is a Vandermonde matrix defined over Fq; consequently, if Ht(j) is considered as a parity-check matrix defining codewords vj, we have here, for each value of j, a Reed-Solomon code, for which decoding algorithms are known which are simple as well as providing good performance;
For example, if a word r has been received, calculation is first made, for j=0, . . . , jmax, of the “aggregate received words”rj=[rj(x1),rj(x2), . . . ,rj(xμ)],in which, for x=x1, x2, . . . , xμ, the “aggregate received words” rj(x) are given by
                                                        r              j                        ⁡                          (              x              )                                =                                    ∑                              p                =                0                                                              λ                  ⁡                                      (                    x                    )                                                  -                1                                      ⁢                                                  ⁢                                                            [                                                            y                      p                                        ⁡                                          (                      x                      )                                                        ]                                j                            ⁢                              r                ⁡                                  (                                      x                    ,                                                                  y                        p                                            ⁡                                              (                        x                        )                                                                              )                                                                    ;                            (        1        )            next use is made of the Berlekamp-Massey algorithm for locating the erroneous symbols of each word rj, followed by the Forney algorithm for the correction of those erroneous symbols, according to the error syndromes vector sj=Ht(j)rjT. Finally, the symbols r(x,yp(x)) are calculated from the symbols rj(x) using the system of equations (1) (or an equivalent system); this system has a unique solution provided that (jmax+1) (the number of equations) is at least equal to λmax, where λmax is greatest of the aggregate cardinals λ(x) (the number of unknowns).
Thus, with respect to the error correction algorithms mentioned previously, the saving in terms of complexity resulting from the use of the method according to application Ser. No. 10/746,144 is significant, despite the necessity to implement an error correction algorithm adapted to Reed-Solomon codes (for example the Forney algorithm) a certain number of times (at least equal to λmax), and to solve for each erroneous aggregate labeled by some value x of X a system of equations (1) of size λ(x); it will be noted however in this connection that λmax is at most equal to a, where a designates, it may be recalled, the exponent of Y in the equation representing the algebraic curve. It will furthermore be noted that the system of equations (1) is a Vandermonde system; as is well-known to the person skilled in the art, the solution of such a system of linear equations is particularly simple.
Furthermore, the implementation of this decoding method is particularly advantageous for a certain type of channel: these are the channels in which the data to transmit are grouped into blocks of predetermined length, and in which the error rate per item of data transmitted is essentially constant within the same block. In other words, such channels are physically characterized in that, most often, the transmission “noises” affect the data by block, and can differently affect different blocks; thus, for certain blocks, the probability of error can be very low or even zero, however for certain other blocks the probability of error could be high and even close to (q−1)/q. This results in a “burst of errors” in terms of symbols of the Galois field.
An example of channels of this type, which is of importance in industrial implementation, is constituted by writing/reading on a hard disk.
This is because the bits composing the symbols of the codewords are usually written by means of a “modulation code” adapted to ensure that certain desirable “spectral” properties are verified, for example the property that the number of 1's, on average, is approximately equal to the number of 0's. To obtain this result, the bits entering the modulator are grouped into blocks a of N bits; depending on the balance between the total number of 1's and the total number of 0's already written, a block α will be written on the disk, either just as it is, or in the form of its complement a′ (where each 1 has been changed into a 0 and vice-versa), such that the new balance is as close as possible to equality.
When a writing/reading error occurs in a particular bit, the resulting ambiguity in the modulation encoding means that it is not possible to individually correct that bit, and that, in practice, it is necessary to accept having to consider the entire block of N bits as erroneous. As it is moreover considered that the bits written and read form strings representing symbols in a Galois field, an erroneous block of N bits will generally encroach on several symbols; for example, if N=48 and q=210, an erroneous block could encroach on 6 strings of 10 successive bits, which results in a burst of errors affecting 6 symbols of the Galois field.
It can thus be seen that the decoding method according to application Ser. No. 10/746,144, which in the first instance corrects erroneous aggregates associated with the received word and not individual erroneous components of that word, is well-adapted to take advantage of such a distribution of the noise on a transmission channel. For this it suffices to insert in an adjacent position in the data stream to be transmitted the components of a (channel) codeword belonging to the same aggregate.
On the other hand, and for the same reason, the number of individual errors which can be corrected with this method may be less than the theoretical error correction capability of the code (as explained above, that theoretical capability is equal to INT[(d−1)/2], where d is the minimum distance of the algebraic geometric code considered). This being the case, the question arises of whether the codes able to benefit from this decoding method all suffer from this drawback to the same extent.